#8159873

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(5,13)(8,14)(10,15)(11,16)(17,18)(19,26)(20,23)(21,25)(22,28)(24,29)(27,31)(30,32)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32)(34,40)(36,41)(42,46)(44,48);(2,7)(3,9)(8,14)(10,15)(17,20,21,29)(18,23,25,24)(19,27,28,32)(22,30,26,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,9)(2,4,7,12)(5,10,13,15)(8,11,14,16)(17,20)(18,24)(19,27)(21,29)(22,31)(23,25)(26,30)(28,32)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,18)(19,22)(20,23)(21,25)(24,29)(26,28)(27,30)(31,32)(33,37,39,35)(34,36,40,41)(38,45,47,43)(42,44,46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,19)(18,22)(20,27)(21,28)(23,30)(24,31)(25,26)(29,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8067069);